When it comes to your eyes and face, patients must be careful when any pathology occurs considering how close everything is situated to the brain. Bacterial infections can easily enter the sinus cavities and will have a pretty easy route to your brain where infections like meningitis can occur. Although this is a "doomsday" situation where a lot of things have to go wrong first, when a patient presents with redness under their eyes, it is something that you need to be mindful of.
There are a few different pathologies possible that can cause this facial redness in the first place. One of the most probable causes of red, swollen skin under your eyes goes by the name of periorbital cellulitis. This is a disease that is caused by bacteria penetrating into the soft tissue of the cheek and will begin its course by causing painful swelling under the eye. Insect bites or sinus infections are two other common causes so there are several different ways to get this infection. If this disease goes untreated, the swelling will only worsen and it can begin to affect the vision and movement of the eyeball. 
If infections are less likely, another possible cause that can lead to swelling under your eyes would be related to your thyroid gland. When your thyroid gland is over-active, you are suffering from a condition known as hyperthyroidism. One of the tell-tale signs of hyperthyroidism is redness and swelling under your eyes. Some other symptoms to watch out for if you think you may have a thyroid problem would be feeling as if you have a racing heart, losing weight even with an increased appetite and feeling extremely warm most of the time. It is a disease that can be easily diagnosed with a simple blood test from your family doctor so don't hesitate if you experience this cascade of symptoms.
Some of the more mild causes of redness under your eyes can be related to changes in atmospheric pressure. Depending on your anatomy, you may have very small sinuses under your eyes. If the cavity is already pretty small, any change in fluid redistribution can lead to this puffiness and redness. Some of the most common complaints from patients would be redness during cold fronts where the atmosphere will drop as bad weather approaches. The best treatment in this situation is to go to your doctor for a simple X-ray or CT scan to rule out any more sinister causes of redness. It is something that you can live with and will usually self-resolve in a number of hours so there is no need for drastic intervention in this case.
The last possible cause of redness under your eyes that I will touch on would be some type of allergic reaction. If you have any allergies to pollen or dust, the first passageway that will be affected will be your nose. Once these particles make contact with the surface of your body, an inflammatory reaction will ensue and patients will have general redness spread across their face. If this is happening to you frequently, it is a good idea to meet with an allergist who will be able to help pinpoint what substances trigger your allergic reactions so you can avoid them in the future.
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